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We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form and give refinements of it. Application on integral mixed means is given.
In the last few decades, the study of classical inequalities, such as the Jensen, the Hölder, the Minkowski and similar inequalities has experienced great expansion. Many extensions, generalizations, improvements, refinements, and applications of these inequalities have been proved to support different research ideas. The first appearance of these inequalities was in integral and discrete forms. However, over the years, the settings in which they were appeared have expanded. For example, classical inequalities have been studied in measure spaces, in Hilbert spaces, on time scale, in families of set-valued mappings, etc. (for instance, see [1,2,3,4,5,6,7,8,9] and references given there). The connection point between numerous generalizations is the theory of isotonic linear functionals. Let us describe this term in detail.
Let E be a non-empty set and L be a linear class of real-valued functions f on E having the properties:
for all ;
, i.e., if for , then .
In this paper we consider a linear functional which is also isotonic, i.e., if and on E then . The basic examples of isotonic linear functionals are sum and R-integral.
Most classical inequalities have a variant involving a linear isotonic functional (see ). Among others, in  (p. 115) we can find the following converse Hölder inequality.
(The converse Hölder inequality for functionals, ). Let L satisfy conditions L1 and L2 and let A be an isotonic linear functional. Let , , and on E with . If for , then
where is a constant defined as
If or , then the reverse inequality in (1) holds provided either or .
Inequality (1) together with the Hölder inequality gives the following chain of inequalities
in which we obtain the lower and the upper bound of the middle term . It is interesting to find more tighter estimation, i.e., to find refinements of the above inequality. In this article, we will focus on finding the refinement of the left side of inequality (3).
Let us say few words about organization of the paper. In Section 2 we give a refinement of the converse Hölder inequality using interpolation result from  (p. 717). Section 3 is devoted to the Beckenbach inequality. Finally, we consider the Minkowski inequality for infinitely many functions and for functionals, state its converse, give refinements of both variants of the converse Minkowski inequality and applied on integral mixed means.
2. Refinement of the Converse Hölder Inequality
The starting point of this consideration is the following interpolating inequality given in  (p. 717). It is proved by using the discrete Jensen inequality and, in fact, this result can be considered as a monotonicity property of the Jensen functional.
If ϕ is a convex function on an interval , , p and q are positive n-tuples such that for all , , , then
Additionally, let us recall the AG inequality in the following form:
(The AG inequality). Let be positive real numbers. If α, β are positive real numbers, such that , then
If or , then the reversed inequality in (5) holds.
The main result of this paper is the following theorem which is a refinement of the known converse Hölder inequality (1). As we will see, its proof is based on the use of a new refinement of the AG inequality. This result has a key role in this paper.
Let A be a linear isotonic functional on a linear class L. Let , and on E with .
Let be, such that for .
If , then
If and , or and , then the reversed inequalities in (6) and (7) hold.
Applying (4) with , , where and are positive real numbers, such that , , , , we obtain the following inequality:
Let h be a function from L, such that for , , and and defined as
Obviously, , . Applying (8) with , , and the above-defined and we obtain
Multiplying that inequality with and using linear functional A it follows
Using formula , replacing h with and k with , where , after multiplying with we obtain
In the following, the term is denoted by .
Applying the AG inequality (5) with , , and we have
Combining (9) and (10) and rearranging, it follows
If , then , and after dividing with we obtain
where is a constant from (2) and is a constant defined as
Since the term is non-negative for inequality (11) is an improvement of the converse Hölder inequality (1).
Let us discuss the other cases for the exponent p.
Let . Then, the function is also convex on , so inequality (9) holds. Additionally, we want to use the AG inequality, but now , and since in this case . So, we have and
Using the above inequality together with (9) and multiplying with we obtain
A term is positive because of the Jensen inequality for a strictly convex function , . After dividing with it follows
In this case, the factor is obviously negative.
Let . Then, is concave on and in (9) reversed sign holds. Using the AG inequality with , , and we obtain
In this case, and dividing above inequality with inequality (13) follows.
Additionally, is obviously negative, so the factor is negative. □
3. Refinement of the Converse Beckenbach Inequality
One of the numerous generalizations of the Hölder inequality is the well-known Beckenbach inequality (). Here we pay attention to the converse Beckenbach inequality. In  the following result (slightly modified) is given.
(The converse Beckenbach inequality, ). Suppose that , and , . Let m and M be positive numbers, such that
where K and N are defined as in Theorem 3 and is defined as in Theorem 5. Dividing the above inequality with and using result (15) we obtain the desired refinement. □
4. The Converse Minkowski Inequality and Its Refinements
In this section, we investigate the converse Minkowski inequality for functionals and the converse of the continuous form of the Minkowski inequality. In  (p. 116), the following converse Minkowski inequality for functionals is obtained.
(The converse Minkowski inequality for functionals, ). Let satisfy assumptions of Theorem 3 with additional property . Let m and M be such that and for .
If , then the second term in the sum on the right-hand side in (18) is non-negative and inequality (18) is a refinement of the known converse (17). Similar proof holds for , . □
The previous investigation does not cover the so-called Minkowski integral inequality. Namely, if and are two measure spaces with sigma-finite measures and , respectively, and if f is a non-negative function on which is integrable with respect to the measure , then for we obtain
The above result is also called “the continuous form of the Minkowski inequality” or “the Minkowski inequality for infinitely many functions” and, for example, it can be found in  (p. 41). Considering the proof of this inequality we can conclude that there exist a related result for other values of the exponent p (see ).
then the reverse inequality holds.
If , the above-mentioned assumptions (20) and the additional one
then the reverse inequality holds.
To our knowledge, in the literature there is no result corresponding to the conversing of the above mentioned results. In the next theorem, we state a converse and a refinement of that variant of the Minkowski inequality. The proof is based on the proof of the continuous form of the Minkowski inequality and on the use of result of Theorem 3.
(The converse continuous form of the Minkowski inequality and refinement). Let and be two measure spaces with sigma-finite measures μ and ν respectively. Let f be a non-negative function on , integrable with respect to the measure .
If for all , then for
where is defined with (2), is defined with (12), and
If with (20) or with (20) and (21), then the reversed inequality holds.
Let us denote
Using Fubini’s theorem we obtain
Using (7) with functional we obtain
Dividing by we obtain inequalities (22) and (23). □
5. Applications on Mixed Means
It is interesting to show how the previously obtained results impact to the study of mixed means.
Let be two positive numbers, . Replacing p by , replacing f by in inequalities (19) and (23), raising to the power and dividing with and , we obtain
where m and M are real numbers, such that .
in inequalities (24) and (25) we obtain the following theorem for mixed means.
Suppose the same conditions as in Theorem 8 hold. If , , then
Additionally, by using (22) the refinement of the above mixed mean inequality can be obtained.
These inequalities are inequalities for mixed means, the second one is a converse of the first inequality. Discrete version of (26) is given in  (p. 109), while its conversion is a new result. It is instructive to calculate mixed means for some special spaces and measures.
Let be such that , , , . If is a non-negative measurable function, then the following inequality holds:
Furthermore, if m and M are real numbers, such that for ,
Using (26) with the following: , , and , , where g is a non-negative measurable function. Then, and .
After substitutions it follows
Replacing in the right-hand side of inequality with the new variable t, we obtain
The same substitution is done in the left-hand side of (30) and we obtain that the left-hand side is equal to:
Replacing with the new variable y in the outer integral, it is equal to
Finally, we obtain
where , , . From (25) with same substitutions we obtain a converse of (28). □
Let us point out that inequality (28) was firstly obtained in  (Theorem 3) and it was used for proving the well-known Hardy inequality. Additionally, let us mention that the above inequalities about mixed means can be refined like as inequalities in previous sections.
In this paper, we have obtained the refinement of the converse Hölder inequality which follows from a monotonicity of the discrete Jensen functional. Additionally, a new conversion of the continuous form of the Minkowski inequality is proved. Our main method in the present work is to use the recently obtained refinement of the converse Hölder inequality in finding refinements of converses of the Beckenbach and the Minkowski inequalities. It would be interesting to explore whether our method can be used to find refinements of other inequalities. Thus, the study of the refinement of other inequalities is a suggested future work.
Writing—original draft, J.P. (Josip Pečarić), J.P. (Jurica Perić) and S.V. All authors have read and agreed to the published version of the manuscript.
The research of first author was supported by the RUDN University Strategic Academic Leadership Program.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
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